Answer is: option2

Solution:

We are given the function:

\[ f(x) = 3 \sin\left(\frac{2x}{3}\right) - 4 \cos\left(\frac{3x}{4}\right) \]

and asked: for \( 0 \leq x \leq 7 \), when is \( f \) increasing most rapidly?

This occurs when the first derivative \( f'(x) \) is maximum.

Differentiate the given function:

\[ f(x) = 3 \sin\left(\frac{2x}{3}\right) - 4 \cos\left(\frac{3x}{4}\right) \]

\[ f'(x) = 3 \cdot \cos\left(\frac{2x}{3}\right) \cdot \frac{2}{3} + 4 \cdot \sin\left(\frac{3x}{4}\right) \cdot \frac{3}{4} \]

\[ f'(x) = 2 \cos\left(\frac{2x}{3}\right) + 3 \sin\left(\frac{3x}{4}\right) \]

Let's evaluate \( f'(x) \) at the answer choices:

• (A) \( x = 0.823 \Rightarrow f'(0.823) \approx 3.440 \)
• (B) \( x = 1.424 \Rightarrow f'(1.424) \approx 3.791 \)
• (C) \( x = 1.571 \Rightarrow f'(1.571) \approx 3.763 \)
• (D) \( x = 3.206 \Rightarrow f'(3.206) \approx 0.943 \)

Maximum value of \( f'(x) \) occurs at \( x = 1.424 \)

You can also graph \( f'(x) \) in desmos and find where the maximum value occurs.